Principled inference for functionals of large structured covariance matrices.

Statistics plays a fundamental role in daily life, allowing costly medical screening, drug development, marketing campaigns or government regulation to be better targeted through improved understanding of the scientific or societal truths underpinning the data we observe. More and more frequently, the scientific truths we wish to learn correspond to a high dimensional parameter. This project considers covariance matrices and related quantities such as inverse covariance matrices, which are particularly important types of high dimensional parameter, arising in numerous statistical applications. When the dimensionality of the covariance matrix is larger than the number of available data points, structure (sparsity in some domain) must be assumed in order to obtain estimates that are well behaved statistically. This project explores new types of structure for covariance and inverse covariance matrix estimation. Some of these structures facilitate uncertainty statements about the true high dimensional parameter rather than simply providing a point estimate. They also allow different estimates to be aggregated without losing statistical accuracy.

The research of this proposal aims to institute a fundamental shift in the cognitive framework underpinning the area of large covariance estimation, which, until now, has only considered point estimators and not set estimators. Large covariance and inverse covariance estimation is an indispensable ingredient to almost every modern statistical procedure, yet the current methodological toolkit is currently insufficiently rich to respond to the demands of practitioners (see Section 2 of the Case for Support document). It also contributes to the area of distributed inference for large scale data. Whilst being widely employed by practitioners, distributed inference is poorly understood theoretically. The research thus contributes to the advancement of statistics as a discipline.

On a national level, improved understanding of the scientific or societal truths underpinning the data we observe allows significant long term economic benefits. For instance, it allows costly medical screening, drug development, marketing campaigns or government regulation to be better targeted.

A direct beneficiary of the research is the Imperial College funded PhD student, who will have the opportunity to work on projects at the heart of modern statistics researched internationally. The organisation who later employs them and the individuals who later work with them will benefit from their expertise in this important area. More generally, the Statistics Section at Imperial College will benefit from exposure to seminar speakers and conference speakers who would not normally have attended (see Justification of Resources document).